What should a husband do with his wife who has asked for a divorce and strayed from Islam, is communicating with a non-Muslim man, and intends to travel without a mahram? And should he encourage her to observe her ‘iddah?

A woman might ask for divorce when she doesn't truly want it, but if it becomes clear that she has started to get to know another man, then this is where the danger lies. It is advisable to be patient with her if she was previously upright in her religion, and to try to prevent her from communicating with this man with gentleness and kindness, and to show affection to her to preserve her as a Muslim, and to try to prevent her from traveling, and to remind her of the companionship between you, and that it is impermissible for her to marry a non-Muslim. And if possible, perform ruqyah (Islamic exorcism/healing). If she responds and repents, then keep her and teach her her religion. However, if she insists on her ways and admonition does not work with her, then divorce her.